It is clear that a basis for a topology generates that topology. (i.e. the smallest topology containing the basis is the topology that the basis is a basis of) But if we look at a generating set for a topology, is this generating set necessarily a basis? I am guessing that no in general (although it is true that the set of "generalized open rectangles" is both a generating set and a basis for the product topology) because otherwise a topology would always be determined by simply the union of its generators. But I have little experience with general topology and since this is just for personnal curiosity, I don't want to waste more time trying to find an example where the generating set is not a basis. Anybody's got an example?